Optimal. Leaf size=30 \[ -\frac {4 x^{3/2}}{3}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \]
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Rubi [A] time = 0.08, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6689, 43} \[ -\frac {4 x^{3/2}}{3}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6689
Rubi steps
\begin {align*} \int \frac {1}{\left (\sqrt {-1+x}+\sqrt {x}\right )^2 \sqrt {-1+x}} \, dx &=\int \left (-\frac {1}{\sqrt {-1+x}}-2 \sqrt {x}+\frac {2 x}{\sqrt {-1+x}}\right ) \, dx\\ &=-2 \sqrt {-1+x}-\frac {4 x^{3/2}}{3}+2 \int \frac {x}{\sqrt {-1+x}} \, dx\\ &=-2 \sqrt {-1+x}-\frac {4 x^{3/2}}{3}+2 \int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx\\ &=2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}-\frac {4 x^{3/2}}{3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 30, normalized size = 1.00 \[ -\frac {4 x^{3/2}}{3}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 18, normalized size = 0.60 \[ \frac {2}{3} \, {\left (2 \, x + 1\right )} \sqrt {x - 1} - \frac {4}{3} \, x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 20, normalized size = 0.67 \[ \frac {4}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, x^{\frac {3}{2}} + 2 \, \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.70 \[ -\frac {4 x^{\frac {3}{2}}}{3}+\frac {4 \left (x -1\right )^{\frac {3}{2}}}{3}+2 \sqrt {x -1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 1} {\left (\sqrt {x - 1} + \sqrt {x}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 21, normalized size = 0.70 \[ \frac {4\,x\,\sqrt {x-1}}{3}+\frac {2\,\sqrt {x-1}}{3}-\frac {4\,x^{3/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.75, size = 53, normalized size = 1.77 \[ - \frac {4 \sqrt {x}}{6 \sqrt {x} \sqrt {x - 1} + 6 x - 3} - \frac {2 \sqrt {x - 1}}{6 \sqrt {x} \sqrt {x - 1} + 6 x - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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